If $g,f:X\rightarrow Y$ are functions and $g$ is monotone then $$ \operatorname{argmin}_{x \in X} f(x) = \operatorname{argmin}_{x \in X} g\circ f(x) . $$

So monotonicity is sufficient for preserving extrema. Are necessary and sufficient conditions known for $g$ to preserve minima? If so does anyone know a good reference?

If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then $$ \operatorname{argmin}_{x \in X} f(x) = \operatorname{argmin}_{x \in X} g\circ f(x) . $$ X and Y are partially ordered sets.

So monotonicity is sufficient for preserving extrema. Are necessary and sufficient conditions known for $g$ to preserve minima? If so does anyone know a good reference?